Find the orthogonal projection of vector v= [ 0 4 0 0] onto the subspace W of R^4 spanned by [-1 -1 1 -1],[1 -1 -1 -1], [1 1 1 -1]. What is projW(vector v)

To find the orthogonal projection of a vector onto a subspace, we need to follow these steps:

1. First, we need to find the basis of the subspace W. In this case, the basis is given by the vectors [-1 -1 1 -1], [1 -1 -1 -1], [1 1 1 -1].

2. Next, we need to find the orthogonal projection of the vector v onto each of these basis vectors. The formula for the orthogonal projection of a vector v onto a vector u is given by:

   proj_u(v) = ((v . u) / ||u||^2) * u

   where "." denotes the dot product, and ||u|| denotes the norm (or length) of the vector u.

3. We then sum up these orthogonal projections to get the orthogonal projection of the vector v onto the subspace W.

Let's calculate:

proj_-1 -1 1 -1 = (([0 4 0 0] . [-1 -1 1 -1]) / ||[-1 -1 1 -1]||^2) * [-1 -1 1 -1] = ((0*-1 + 4*-1 + 01 + 0-1) / (1^2 + 1^2 + 1^2 + 1^2)) * [-1 -1 1 -1] = (-4 / 4) * [-1 -1 1 -1] = -1 * [-1 -1 1 -1] = [1 1 -1 1]

proj_1 -1 -1 -1 = (([0 4 0 0] . [1 -1 -1 -1]) / ||[1 -1 -1 -1]||^2) * [1 -1 -1 -1] = ((01 + 4-1 + 0*-1 + 0*-1) / (1^2 + 1^2 + 1^2 + 1^2)) * [1 -1 -1 -1] = (-4 / 4) * [1 -1 -1 -1] = -1 * [1 -1 -1 -1] = [-1 1 1 1]

proj_1 1 1 -1 = (([0 4 0 0] . [1 1 1 -1]) / ||[1 1 1 -1]||^2) * [1 1 1 -1] = ((01 + 41 + 01 + 0-1) / (1^2 + 1^2 + 1^2 + 1^2)) * [1 1 1 -1] = (4 / 4) * [1 1 1 -1] = 1 * [1 1 1 -1] = [1 1 1 -1]

Finally, we add these projections together to get the orthogonal projection of v onto W:

proj_W(v) = proj_-1 -1 1 -1 + proj_1 -1 -1 -1 + proj_1 1 1 -1 = [1 1 -1 1] + [-1 1 1 1] + [1 1 1 -1] = [1 3 1 1]

So, the orthogonal projection of the vector v onto the subspace W is [1 3 1 1].